Transport of passive, dissolved compounds in fully-saturated complex porous media frequently exhibits non-Fickian characteristics. One of the most interesting questions is to ascertain the time scales at which it is possible to
describe transport as a statistically independent process. Therefore we study
the mechanisms for evolution and then the decrease of non-Fickianity as a
function of increasing time. Adopting the Lagrangian perspective, we provide a non-linear copula analysis of advective-diffusive processes by analyzing particle trajectories in a real porous media, as provided by direct numerical simulations on the three-dimensional image of Doddington sandstone.
First, we analyze the memory effects between time-consecutive particle position increments and cross-dependence between longitudinal and transver
sal particle position increments as a function of given time increments and
time lags between consecutive time increments. Second, we investigate the
influence of the P´eclet regime on the temporal evolution of dependence. Our
main findings are: (a) Cross-dependence between longitudinal and transver
sal particle position increments is persistent over the investigated range of
time increments, even though this aspect has been neglected up to date. (b)
Lower P´eclet numbers lead to a weaker dependence that is, however, more
persistent over time than in higher-P´eclet transport regimes. We confirm that
non-Fickianity comes from spatial coherence associated with heterogeneities
of the velocity field that introduce cross-dependence and memory into the
transport process. Overall, we show that memory and cross-dependence are persistent in and among all directions, that the dependence is highly-nonlinear,
occurs at different temporal scales, and is dependent on the P´eclet number.